TITLE: Check if $\det(I + S) = 1 + \operatorname{trace}(S)$ holds ?
QUESTION [15 upvotes]: I saw the following statement in my homework and we are asked to make use of the statement: 
If $S$ is a symmetric matrix then 
$$\det(I + S ) = 1 + \operatorname{trace}(S).$$
However, I am not sure if this really holds. Please give me a hand here please

REPLY [4 votes]: Holds if $S$ is of rank $1$. A simple case for that is when $S=a\cdot b^T$ where $a$ and $b$ are column vectors.